Irrational?

We show that any such $s$ must be rational.

Note that $0.\underbrace{0\ldots 0}_{n\text{ }0\text{s}}\overline{7} = \frac{7}{9\cdot10^n}$ is a rational number. If $s$ has a finite number of non-seven digits, there must be a rightmost non-seven digit. Say this occurs at the $n$ th decimal place. Then, every decimal place to the right of the $n$ th place must be a seven. So we can write $s=k+0.\underbrace{0\ldots 0}_{n\text{ }0\text{s}}\overline{7} = k+\frac{7}{9\cdot10^n}$ where $k<1$ has $n$ decimal digits. Since the decimal representation of $k$ terminates, it is a rational number and, as the sum of rationals, $s$ must also be rational.